5. Application to GRS 1915+105
The number of parameters in the model outlined above is large. In order to reduce this number we assume that the jets in microquasars are freely expanding, i.e. , and that the flux of magnetic energy through the jet is conserved, i.e. . Since we assume the bulk velocity of the jet material to be constant, this implies that the ratio of the kinetic energy and the energy of the magnetic field is constant as well. Furthermore, we impose symmetry between the approaching and receding sides of the source in the sense that the model parameters describing the jet are the same on both sides. This may be a poor assumption as the jets of GRO J1655-40 are observed to be asymmetric (Hjellming & Rupen 1995). The model then depends on five free parameters: The e-folding distance of the number of relativistic particles within the jet accelerated by the shock, , the bulk velocity of the shocked jet material, , the maximum Lorentz factor up to which relativistic particles are initially accelerated, , the slope of the initial power law energy spectrum of these particles, p, and the energy density of the magnetic field at , . The acceleration rate of relativistic particles at the normalisation radius, , is in principle also a free parameter. However, from Eq.( 15) we note that it is only a multiplicative factor in the calculation of the total radio emission of the jet. We therefore use it to normalise the model in such a way that for a given set of model parameters is such that the difference between the model predictions and the observational data is smallest for this given set of parameters.
Many radio outbursts of a number of microquasars have been observed. But to constrain the model parameters in a meaningful way, we would ideally need radio observations at two or more frequencies which clearly resolve the approaching and the receding jet component. Furthermore, the resolution should be sufficient to decide whether one of these jet components consists of multiple subcomponents, i.e. multiple shocks, the emission of which may be blended in observations of lower resolution. To date there are very few simultaneous multi-frequency observations which come even close to this ideal situation. The best studied radio outburst of any microquasar is still that of March 19th 1994 of GRS 1915+105 (Mirabel & Rodríguez 1994). This is also the outburst studied by Atoyan & Aharonian (1999). To test our model we will use the comparatively large data base accumulated during this event.
5.1. The observations
The radio outburst of GRS 1915+105 which occurred in March 1994 was one of the strongest recorded for this object. The flux density at 1.4 GHz exceeded 1 Jy which is at least ten times higher than the radio flux in quiescence (Rodríguez et al. 1995). The outburst was observed with the VLA in A-array at 8.4 GHz during 7 epochs covering almost 42 days. Except for the first of these, the two jet components were resolved at this frequency (Mirabel & Rodríguez 1994). For the first unresolved and one further epoch measurements with the VLA are also available at 4.9 GHz and 15 GHz. In addition, GRS 1915+105 was monitored during this time by the Nancay telescope at 1.4 GHz and 3.3 GHz (Rodríguez et al. 1995). There are 24 flux measurements at each frequency but the source is unresolved at these frequencies. After a reconfiguration of the VLA four more observations of GRS 1915+105 of lower resolution were obtained at 8.4 GHz in B-array (Rodríguez & Mirabel 1999). These measurements cover roughly another 28 days but there is a gap of about 40 days between the end of the A-array observations and the start of the B-array campaign.
From the resolved VLA observations at 8.4 GHz Mirabel & Rodríguez (1994) determined a velocity for the jet components of 0.92 c and an angle of the jets to the line of sight of 70o. This assumes that the approaching and the receding component travel at the same velocity in opposite directions. Furthermore, it is assumed that GRS 1915+105 is located 12.5 kpc away from us. Fender et al. (1999) observed another radio outburst of GRS 1915+105 in October 1997 and found a higher intrinsic velocity which is inconsistent with a distance of 12.5 kpc. They argue that the most likely distance for this object is 11 kpc which then implies that the velocity of the jet components in March 1994 was 0.86 c and the angle of the jets to the line of sight is 68o. In the following we will adopt these later values. Extrapolating back the trajectories of the two jet components Mirabel & Rodríguez (1994) find that the outburst started at 20 hours on March 19.
Figs. 1 and 2 show all available flux density measurements as a function of time. Rodríguez & Mirabel (1999) note that another outburst of GRS 1915+105 occurred on April 21. The jet components of this new outburst are clearly visible as an unresolved emission peak coincident with the source centre in the VLA radio map of epoch 6. The approaching component of this new outburst is also distinctly visible on the map of the following observing epoch while the receding component is probably blended with that of the previous outburst of March 19. The signature of this later outburst as a sudden increase of the radio flux is not very distinct at 1.4 GHz and 3.3 GHz. Even at 8.4 GHz the situation in terms of the total flux density is somewhat unclear. However, the outburst can be easily identified as a separate event from the one of March 19 because the VLA maps reveal an emission peak distinct from those of the earlier outburst. The later observation epochs at the VLA with lower resolution detect the jet components of the April 21 burst while the components of the March 19 event were not detected (Rodríguez & Mirabel 1999). This is rather puzzling as the extrapolation of their lightcurves from the earlier observations indicate that they should still have been visible during these later observations.
5.1.1. The blending of outbursts
A close examination of the total radio flux at 8.4 GHz reveals another sudden increase around observation epoch 4 (April 9, Fig. 1). This increase is also detected at 1.4 GHz and 3.3 GHz by the Nancay telescope on April 5 (Rodríguez et al. 1995, Fig. 2). Atoyan & Aharonian (1999) point out that this may be caused by a sudden additional injection of fresh relativistic electrons in the `blobs' of gas they consider in their model. An alternative interpretation, which we will adopt here is that another, smaller outburst occurred shortly after April 4. In this case, the radio emission caused by the propagation of a new shock down the jet is most likely blended with that of the earlier event of March 19 because of the limited resolution of the observations. If the smaller outburst occurred at April 5, 0:0 hours, and assuming that the apparent shock velocity of this outburst is equal to that of the March 19 event, i.e. 17.5 mas day-1 (Rodríguez & Mirabel 1999), then the distance of the shock from the source centre on the approaching jet side at observing epoch 4 would be roughly 0.1". The position of the emission peak on April 9 is given as 0.36" from the source centre (Mirabel & Rodríguez 1994). The resolution of the VLA in A-array at 8.4 GHz is 0.3" and this means that a secondary peak caused by a later, somewhat weaker outburst as proposed here could not be detected as an individual structure. The situation on the receding jet side with its lower expansion velocities is even worse. However, the radio emission of the additional outburst would contribute to the total radio flux of the source and we believe that this has been detected here.
In summary, we will assume that the observations outlined above cover three separate radio outbursts of GRS 1915+105. The first and strongest occurred on March 19. The jet components of this burst are detected until the end of the first set of VLA observations (April 30). By the time of the second VLA campaign (June 13) they had vanished although the extrapolation of their earlier lightcurves suggested that they should still be observable. The second much weaker outburst occurred shortly after April 4, since the Nancay data show a sudden increase in radio flux on April 5 but not on April 4. The radio emission of this event is most likely blended with that of the first outburst and there is no sign of this second burst in the second set of VLA observations. The third outburst finally started on April 21 and was intermediate in strength. The radio emission caused by this event is clearly detected in the source centre on April 23 and the approaching jet component can be seen in the VLA map obtained on April 30. Both jet components are clearly detected in all four VLA observing epochs in June and July.
5.1.2. Data used in the modeling
In the continuous jet model for microquasars outlined above, only one shock is thought to travel outwards in each jet. Because of this, only observational data from observing epochs during which we can be sure that there was only one shock per jet contributing to the radio emission can be used in constraining the model parameters. From the above discussion it is clear that we can only use the three unresolved VLA measurements at 4.9 GHz, 8.4 GHz and 14.9 GHz from March 24 and the two following resolved observations at 8.4 GHz. These later observation provide us with separate flux measurements for the approaching and receding jet components. We also use the measurement of the receding jet component of April 9, since it seems likely that all the flux of the second outburst was attributed to the approaching component by Mirabel & Rodríguez (1994). Alternatively, this measurement can be taken as an upper limit. Of the Nancay observations we use all flux measurements starting March 24 through to April 4. There are eight observations during this time at 1.4 GHz and 3.3 GHz. For all 24 measurements used to constrain the model we assume the conservative error of 46% suggested by Rodriguez & Mirabel (1999) as opposed to the original error of 5% quoted by Rodríguez et al. (1995). In practice we did not use the eight 3.3 GHz data points as their inclusion led to significantly worse fits of the model to the observational data. See the next section for a discussion of this point.
5.2. Constraining the model parameters
We use the model outlined above to calculate the expected radio flux at the times GRS 1915+105 was observed during the outburst starting March 19. The `goodness of fit' of the model for a given set of free parameters to the observations was assessed by calculating the sum of the -differences at the times the source was observed. The best-fitting model was then found by minimising this -value using a 4-dimensional downhill simplex method (Press et al. 1992). The remaining fifth model parameter, the energy density of the magnetic field at the normalisation radius, , was set `by hand' to five different values. The normalisation radius, , was set to m which corresponds to the unprojected distance of the jet shock from the source centre at the time of the first VLA observation, i.e. March 24.
The results of the model fits are summarised in Table 1. The model fits the observational data equally well for all five adopted values for the strength of the magnetic field. As expected from Eq. (16) we find that the maximum Lorentz factor up to which relativistic particles are accelerated correlates strongly with the value of the energy density of the magnetic field. The values found for in the model fits follow almost exactly a law. This shows that the model presented here cannot be used to constrain the strength of the magnetic field within the jet independently of the maximum energy of the relativistic particles. To proceed we adopt in the following as our fiducial model. This corresponds to the equipartition value of the magnetic field of T (0.1 G) found by Atoyan & Aharonian (1999) for the chosen value of . Note however, that the energy density of the magnetic field and that of the relativistic particles does not stay in equipartition for all times in our model.
Table 1. Results of the model fits. The first five fits are obtained without the 3.3 GHz data points. is the energy density of the magnetic field, is the radius beyond which the acceleration rate of relativistic particles decreases exponentially in units of (see Eq. 6), is the bulk velocity of the shocked jet material, is the initial high energy cut-off of the energy spectrum of the relativistic particles, p is the initial slope of this spectrum and is its normalisation.
Figs. 1 and 2 show the model predictions of the fiducial model compared to the observational data. Also shown are the predictions of the model for the best-fitting parameters when using also the 3.3 GHz data. If we use all available data, the model predictions at the higher observing frequencies are rather low at early times. For the VLA measurements at 4.9 GHz and 14.9 GHz (not shown in Figs. 1 and 2) on March 24 the model predicts flux densities of 759 mJy and 385 mJy respectively. This is much lower than the measured flux densities of 887 mJy and 514 mJy at these frequencies. A closer inspection of Fig. 2 also shows that the measurement at 4.9 GHz actually exceeds those taken at 3.3 GHz at comparable times, which is unlikely to be real. Table 1 also shows that the fit obtained including the 3.3 GHz data is much worse in terms of the reduced -values than that excluding them. Also the flux densities predicted by our fiducial model which excludes the 3.3 GHz data, 926 mJy at 4.9 Ghz and 499 mJy at 14.9 Ghz, are much closer to the observations. Finally, the predictions of this model at 3.3 GHz fit the observations well at this frequency apart from the early observing epochs (see Fig. 2). We therefore believe that the model parameters found using our fiducial model and excluding the 3.3 GHz data are more reliable than those found when including these additional measurements.
To estimate the expected error of the model parameters of our fiducial model we calculated the -value for a large set of combinations of the 4 free model parameters. The uncertainties quoted in Table 1 are 1- errors corresponding to those parameter ranges for which . Note that the uncertainties of the model parameters, particularly those of and p, are large while the light curves predicted by the model pass the data points well within the error bars of the flux measurements (see Figs. 1 and 2). This suggests that the quoted errors of the observed fluxes, at least for the VLA data points, are too conservative which also results in an overestimation of the uncertainties of the model parameters.
5.3. Comparison with the later observations
The exponential function which describes the change in the acceleration rate of relativistic particles as a function of the position of the shock in the jet, Eq. (6), implies that the radio flux caused by the first outburst on March 19 decreases quickly once exceeds . This effect is clearly visible in Fig. 1 at 8.4 GHz. The model predicts that without the additional blended radio emission caused by the second and third outbursts around April 5 and April 21 the two jet components would have faded much more rapidly than is observed. This effect is less pronounced at lower frequencies (see Fig. 2). However, even for these the predicted and the observed light curves steepen somewhat roughly 15 days after the start of the first outburst. The continued steepening of the lightcurves of the two jet components at 8.4 GHz can also explain why they were not detected during the second observing campaign at the VLA (see Fig. 1).
We tried replacing the exponential in Eq. (6) with a simple power law. The data can be adequately fitted with this modified model as well. However, this change in the temporal behaviour of also leads to a much increased flux at low observing frequencies at later times. Using this modified model we found a flux at 1.4 GHz 40 days after the start of the first outburst exceeding the observations by a factor of at least 1.3 even without considering the possible contribution from later outbursts. This supports the picture of two colliding shells of jet material of finite width causing the internal shock. The rate at which energy is dissipated is roughly constant during the collision and decreases rapidly once the two shells have merged.
Comparing the model lightcurves with the observational data the signature of the second outburst starting around April 5, about 16 days after the start of the first outburst, can clearly be detected at 8.4 GHz. The increase in the radio emission caused by the second event is less dramatic at 3.3 GHz and 1.4 GHz but can still be seen in Fig. 2. The third outburst of April 21, 32 days after the first burst, is seen as excess emission at 8.4 GHz and 3.3 GHz but is less obvious at 1.4 GHz. We note that in general the smooth lightcurve predicted by our model fits the VLA observations much better than the flux measurements at 1.4 GHz and 3.3 GHz taken with the Nancay telescope. This may imply larger errors for the low frequency data which hide to some extent the signatures of the second and third outburst which are much weaker than the first.
© European Southern Observatory (ESO) 2000
Online publication: April 17, 2000